Offers an intuition about directional derivatives. Some of the graphics were made more attractive by code borrowed from the Calculus & Mathematica program in the Department of Mathematics at the University of Illinois at Urbana-Champaign.

See also The Gradient Vector.

We start with a surface, in this example and a point .

If we stood at that point, there are infinitely many directions in which we could walk. Each direction has a certain steepness associated with it. This is the issue we want to deal with here. First let's take a closer look at the surface:

Walking in a certain direction means that we head along a vector. Here we show the vector (in blue), in the xy-plane, with initial point at , from two different viewpoints.

This vector is contained in a vertical plane.

The intersection of the plane with the surface is a curve (trace). It is this curve that we would walk along if we began at the point and walked in the direction of the vector . The curve has a tangent line at the point .

Now we define the "directional derivative of f at (a,b) in the direction of u", where is unit vector, as: This number (you can think of it as the "slope" of the above tangent line) tells us the "steepness" of the surface in this direction. In this case .

This work is part of the Multivariable Calculus with Mathematica Project at the University of Oklahoma.